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Graphical Vector Addition
'Graphical Vector Addition' What are "vectors," anyways? Vectors are measurements that have both magnitude and direction. Vector quantities are used frequently in both math and physics; vector usage has both basic and complex applications. Vectors are represented by straight lines with a head on one end (denoted by an arrow,) and a tail (denoted by a “flat” end) on the other. As might have been assumed, the arrow represents the direction of the vector, and the length of the line represents the distance of the vector. meaning of the word "vector" itself has a definition that lends itself to desciribing many other phyics related terms (each that complys with having both a direction and a magnitude) such as velocity as opposed to speed, which connotes only magnitude and not direction. The history of Graphical Vectors Vectors come from the manipulation of the long standing theorem made by Pythagoras during the early years of mathematical foundation as well as the methods of Cartesian geometry found in Bolzaon's work in the early 19th century. According to some, the Pythagorean Theorem is one of the greatest contributions to the mathematical and scientific community; it was also one of the earliest known theorems known to ancient civilizations. The actual date of Pythagoras’ discovery of the notion that the sum of the squares of a right triangle equals the hypotenuse is uncertain, yet many say that it came about around the time that he founded the Pythagorean School of Mathematics (500 BC). The basis of vector operations lie within the concept that (a2 + b2 = c2) assuming that “a” and “b” are the legs of a triangle where the angle between them is 90 degrees, and that “c” is the hypotenuse. Mathematicians soon realized that the components that make up Pythagora's special triangles could be rearranged in hopes of being added; this "sum" is called the resultant. Bolzano's work delt specifically with the concept that points, lines and planes are undefined elements and can be operated on as distinct particles. This idea had a strong influence on the concept of linear space and visual abstraction. Bolzano was interested in creating concrete objects from spacial ones; he called vectors "abstract linear operators" that were limited to the coordinate plane only. With the ideas from both Bolzano and Pythagora, the mathematical and scientific world was forever changed; and the resulting technology was directly influenced. Although Bolzano inspired many works from genius' such as Poncelet and Chasles, the notion of vector addition has only been recently accepted, for many years the concept of vectors was murky and there was much confusion. However, early into the 20th century there was a rise of interest in vector analysis and its relation to architecture. say that this may have been a result of the rise of technology and the need to understand two-dimensional figures before composing three-dimensonal ones. If you're interested, you can take a look at "The Evolution of the Idea of a Vectorial System" written by Michael J. Crowe. Who else uses the concept of Graphical Vector Addition? Just as the Pythagorean Theorem is internationally famous, so too is graphical vector addition. Graphical vector addition has been around for centuries, thus it is not suprising that numerous countries utilize its functions and apply it to contemporary society (both in the real world and the simulated-school- world). In common society, most cultures use vector addition in very similar ways, and for very similar reasons. Either to solve for unknown distances, describe uniform motion, or create basic skeleton diagrams for 3-dimensional objects. For example, in modern society, students who are studying to become engineers use vector applications frequently as means of simplifying the visualiztion of 3-dimensional geometry, or calculating the area of a potential pyramid. However, a hundred years ago or so, vectors had various primary purposes depending on the culture it was used in. For example, in Ireland in the 1830's vectors were used as evidence to supporty the thoery of complex numbers, and their position on the plane. Hamilton, a devoted Irish mathematician, used vectors to represent complex numbers in a two-dimensional space, and attempted to add a third and fourth vector in hopes of finding a location for both the complex and imaginary numbers. 'Working with Vectors' Vector Addition: part 1, the basics The simplest type of a vector operation is the addition of vectors in order to determine the resultant. —The resultant can also be referred to as the "resultant displacement" of an object, which means the distance the object is from its original position after every tranformation it undergoes.— times students are familiar with a little bit of previous knowledge concerning the idea that the net (total) force on an object is the vector sum of all the individual forces acting on that object. Assuming that the student is completely unfamiliar with net forces, etc. lets start at the beginning. We first denote one direction as positive and the other as negative. Lets assume that right is positive, and left is negative. Next we draw a vector pointing in a specific direction and give it a defined magnitude. Lets assume that the vector points to the right (positive direction) with a magnitude of 5. javascript:insertTags(' This means that any vector also pointing towards the right can be added to the first vector, regardless of the magnitude. Lets propose that this second vector has a magnitude of 9 and also points toward the right. These two vectors will be added as positive integers: 5 + 9 = 14. The resultant vector is 14. Now, when one vector points in one direction, and the other vector points in the opposite direction, we assume that one is a positive vector (as it points in the positive direction) and the other is a negative vector (as it points in the negative direction); thus adding these two vectors becomes more like a subtraction problem. Lets suppose that the vector pointing towards the right has a magnitude of 4, and the vector pointing towards the left has a magnitude of 7. In order to add these two vectors we must add positive 4 and negative 7, which looks like: 4 + (-7) = 4 — 7 = (-3). If two vectors are equal in magnitude, but opposite in direction, then they will cancel eachother out and the resultant vector will be zero. Before we try to do some problems on our own let's review the steps for properly adding vectors. 1. Label one direction positive (+) and the other direction negative (-). 2. Draw proper vectors, preferably to scale. 3. Position vectors so that their heads point in their respecatble directions. 4. Add them together using the basic laws of arithmetic. 5. Convert pos/neg vectors back into proper directions Regents Review Sample Problems These questions were taken from the Barron's Regents Review Book question: What is the resultant of 5N pointing north and 18N pointing south? (Don't forget to draw a diagram and label each part!) solution: Lets say that north is positive, and south is negative. If so our diagram looks like this: To add the two vectors we have 5N + (-18N) = (-13N). Thus our answer is 13N pointing south. question: A brid lifes north 3 kilmoeters and then south 4 kilometers, what is the resultant displacement of the bird? solution: If we say that north is negative, and south is positive then we have a negative vector of magnitude 3km and a positive vector of magnitude 4km. The resultant is what we get when we add the two vectors: (-3km) + 4km = 1km. The resultant displacement of the bird is 1km. question: Suppose a child walking home from school walks 2blocks east, until he realizes that he left his textbook at school and decides to walk the 2 blocks back and then walk to his house 8 blocks from school. How great is his resultant displacement from the point at which he realizes that he left his book at school? solution: Lets say that east is positive and west is negative. The kid walks 2 positive blocks, 2 negative blocks, and then 8 more positive blocks. Except, that the question specifically asks for the childs resultant distance from the point 2 blocks away from school. This means that in reality, we only have 2negative blocks and 8positive blocks to add together. Thus, our vector addition looks like this: (-2) + 8 = 6. The child has a total resultant displacement of 6 blocks. question: Two forces of 5 newtons and 15 newtos acting concurrently could have a resultant with a magnitude of 1) 5N 2) 10N 3) 25N 4) 75N solution: The only answer here that complys with the properties of vectors is choice (2) 10N. This is because two forces of 5N and 15N can either be added where both 5N and 15N are positive vectors, or with 5N as the positive vector, and 15N as the negative vector, or 5N as the negative vector and 15N as the positive vector. Here are what the two possibilites look like: 5N + (-15N) = (-10N), which means 10N in the opposite direction from that of the original vector; or (-5N) + 15N = 10N, once again where the resultant vector is in the opposite direction of that of the original vector; the last possible answer would be 5N + 15N = 20N pointing in the same direction as the original vector. The only choice that fits these requirements, is 10N. Since the answer choices don't specify direction, we say that it is negligible, and thus 10N is the answer. question: If a woman runs 100 meters north and then 70 meters south, her total displacement will be 1) 30m N 2) 30m S 3) 170m N 4) 170m S solution: Lets say that north is positive and that south is negative. Now we can write an equation that looks like this: 100m + (-70m) = 30m. Since our answer is positive, we know that the resultant vector must be pointing north. Thus, our answer is 30m N.